Properties

Degree $2$
Conductor $960$
Sign $0.447 + 0.894i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (2 − i)5-s − 2i·7-s − 9-s − 2·11-s − 6i·13-s + (1 + 2i)15-s − 2i·17-s + 2·21-s − 4i·23-s + (3 − 4i)25-s i·27-s − 8·31-s − 2i·33-s + (−2 − 4i)35-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 0.603·11-s − 1.66i·13-s + (0.258 + 0.516i)15-s − 0.485i·17-s + 0.436·21-s − 0.834i·23-s + (0.600 − 0.800i)25-s − 0.192i·27-s − 1.43·31-s − 0.348i·33-s + (−0.338 − 0.676i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.447 + 0.894i$
Motivic weight: \(1\)
Character: $\chi_{960} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33864 - 0.827325i\)
\(L(\frac12)\) \(\approx\) \(1.33864 - 0.827325i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 + (-2 + i)T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00753669308419175057550088676, −9.193411808798071220306016277263, −8.254969611591929950200229108632, −7.48323700282678448355579266359, −6.27774383454896964527410242383, −5.37964832598748357281358339416, −4.77142167373993736883932077938, −3.50680404998244847127583535008, −2.43558900516617993360117622150, −0.72102067387008897618786772945, 1.76623382470639652667722352824, 2.41364340619497992275745377862, 3.77205334529209662907691509218, 5.24833189380140533966738785239, 5.88131568020044699660890229835, 6.78755993818593172158689370167, 7.46062972812106738853262156018, 8.715158220990760030641822257168, 9.212099851975808375776571483064, 10.12096110965081070063231957866

Graph of the $Z$-function along the critical line